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The ability to detect and adapt to changes in data distributions is crucial to maintain the accuracy and reliability of machine learning models. Detection is generally approached by observing the drift of model performance from a global point of view. However, drifts occurring in (fine-grained) data subgroups may go unnoticed when monitoring global drift. We take a different perspective, and introduce methods for observing drift at the finer granularity of subgroups. Relevant data subgroups are identified during training and monitored efficiently throughout the model's life. Performance drifts in any subgroup are detected, quantified and characterized so as to provide an interpretable summary of the model behavior over time. Experimental results confirm that our subgroup-level drift analysis identifies drifts that do not show at the (coarser) global dataset level. The proposed approach provides a valuable tool for monitoring model performance in dynamic real-world applications, offering insights into the evolving nature of data and ultimately contributing to more robust and adaptive models., Comment: Currently under submission

This paper provides a detailed analysis of the Dirichlet boundary value problem for linear elliptic equations in divergence form with $L^p$-general drifts, where $p \in (d, \infty)$, and non-negative $L^1$-zero-order terms. Specifically, by transforming the general drifts into weak divergence-free drifts, we establish the existence and uniqueness of a bounded weak solution, showing that the zero-order term does not influence the quantity of the unique weak solution. Additionally, by imposing the $ VMO$ condition and mild differentiability on the diffusion coefficients and assuming an $L^s$-zero-order terms with $s \in (1, \infty)$, we demonstrate the existence and uniqueness of a strong solution for the corresponding non-divergence type equations. An important feature of this paper is that, due to the weak divergence-free property of the drifts in the transformed equations, the constants appearing in our estimates can be explicitly calculated, which is expected to offer significant applications in error analysis., Comment: The first version, 27 pages

Accurate modeling of tidal interactions is crucial for interpreting recent JWST observations of the thermal emissions of TRAPPIST-1~b and c and for characterizing the surface conditions and potential habitability of the other planets in the system. Indeed, the rotation state of the planets, driven by tidal forces, significantly influences the heat redistribution regime. Due to their proximity to their host star and the estimated age of the system, the TRAPPIST-1 planets are commonly assumed to be in a synchronization state. In this work, we present the recent implementation of the co-planar tidal torque and forces equations within the formalism of Kaula in the N-body code Posidonius. This enables us to explore the hypothesis of synchronization using a tidal model well suited to rocky planets. We studied the rotational state of each planet by taking into account their multi-layer internal structure computed with the code Burnman. Simulations show that the TRAPPIST-1 planets are not perfectly synchronized but oscillate around the synchronization state. Planet-planet interactions lead to strong variations on the mean motion and tides fail to keep the spin synchronized with respect to the mean motion. As a result, the sub-stellar point of each planet experiences short oscillations and long-timescale drifts that lead the planets to achieve a synodic day with periods varying from $55$~years to $290$~years depending on the planet., Comment: 7 pages, 6 figures

One proves the well-posedness in the Sobolev space H^{-1} of nonlinear Fokker-Planck equations with singular drifts.Applications to existence of strong solutions to McKean-Vlasov equations are given.

With the boom of machine learning (ML) techniques, software practitioners build ML systems to process the massive volume of streaming data for diverse software engineering tasks such as failure prediction in AIOps. Trained using historical data, such ML models encounter performance degradation caused by concept drift, i.e., data and inter-relationship (concept) changes between training and production. It is essential to use concept rift detection to monitor the deployed ML models and re-train the ML models when needed. In this work, we explore applying state-of-the-art (SOTA) concept drift detection techniques on synthetic and real-world datasets in an industrial setting. Such an industrial setting requires minimal manual effort in labeling and maximal generality in ML model architecture. We find that current SOTA semi-supervised methods not only require significant labeling effort but also only work for certain types of ML models. To overcome such limitations, we propose a novel model-agnostic technique (CDSeer) for detecting concept drift. Our evaluation shows that CDSeer has better precision and recall compared to the state-of-the-art while requiring significantly less manual labeling. We demonstrate the effectiveness of CDSeer at concept drift detection by evaluating it on eight datasets from different domains and use cases. Results from internal deployment of CDSeer on an industrial proprietary dataset show a 57.1% improvement in precision while using 99% fewer labels compared to the SOTA concept drift detection method. The performance is also comparable to the supervised concept drift detection method, which requires 100% of the data to be labeled. The improved performance and ease of adoption of CDSeer are valuable in making ML systems more reliable.

In this paper, we investigate the convergence rate of the averaging principle for stochastic differential equations (SDEs) with $\beta$-H\"older drift driven by $\alpha$-stable processes. More specifically, we first derive the Schauder estimate for nonlocal partial differential equations (PDEs) associated with the aforementioned SDEs, within the framework of Besov-H\"older spaces. Then we consider the case where $(\alpha,\beta)\in(0,2)\times(1-\tfrac{\alpha}{2},1)$. Using the Schauder estimate, we establish the strong convergence rate for the averaging principle. In particular, under suitable conditions we obtain the optimal rate of strong convergence when $(\alpha,\beta)\in(\tfrac{2}{3},1]\times(2-\tfrac{3\alpha}{2},1)\cup(1,2)\times(\tfrac{\alpha}{2},1)$. Furthermore, when $(\alpha,\beta)\in(0,1]\times(1-\alpha,1-\tfrac{\alpha}{2}]\cup(1,2)\times(\tfrac{1-\alpha}{2},1-\tfrac{\alpha}{2}]$, we show the convergence of the martingale solutions of original systems to that of the averaged equation. When $\alpha\in(1,2)$, the drift can be a distribution., Comment: 30 pages

This article aims at quantifying the long time behavior of solutions of mean field PDE systems arising in the theory of Mean Field Games and McKean-Vlasov control. Our main contribution is to show well-posedness of the ergodic problem and the exponential turnpike property of dynamic optimizers, which implies exponential convergence to equilibrium for both optimal states and controls to their ergodic counterparts. In contrast with previous works that require some version of the Lasry-Lions monotonicity condition, our main assumption is a weak form of asymptotic monotonicity on the drift of the controlled dynamics and some basic regularity and smallness conditions on the interaction terms. Our proof strategy is probabilistic and based on the construction of contractive couplings between controlled processes and forward-backward stochastic differential equations. The flexibility of the coupling approach allows us to cover several interesting situations. For example, we do not need to restrict ourselves to compact domains and can work on the whole space $\mathbb{R}^d$, we can cover the case of non-constant diffusion coefficients and we can sometimes show turnpike estimates for the hessians of solutions to the backward equation., Comment: 58 pages

In this paper, long time quantitative propagation of chaos in $L^1$-Wasserstein distance for mean field interacting particle system is derived, where the diffusion coefficient is allowed to be interacting and the drift is assumed to be partially dissipative. The main tool relies on reflection coupling, the gradient estimate of the decoupled SDEs, and the Duhamel formula for two semigroups associated to two time-inhomogeneous diffusion processes on $(\mathbb{R}^d)^N$. Moreover, the long time quantitative propagation of chaos in $L^\eta$($\eta\in(0,1)$)-Wasserstein distance is also obtained., Comment: 19 pages

User experience in data visualization is typically assessed through post-viewing self-reports, but these overlook the dynamic cognitive processes during interaction. This study explores the use of mind wandering -- a phenomenon where attention spontaneously shifts from a primary task to internal, task-related thoughts or unrelated distractions -- as a dynamic measure during visualization exploration. Participants reported mind wandering while viewing visualizations from a pre-labeled visualization database and then provided quantitative ratings of trust, engagement, and design quality, along with qualitative descriptions and short-term/long-term recall assessments. Results show that mind wandering negatively affects short-term visualization recall and various post-viewing measures, particularly for visualizations with little text annotation. Further, the type of mind wandering impacts engagement and emotional response. Mind wandering also functions as an intermediate process linking visualization design elements to post-viewing measures, influencing how viewers engage with and interpret visual information over time. Overall, this research underscores the importance of incorporating mind wandering as a dynamic measure in visualization design and evaluation, offering novel avenues for enhancing user engagement and comprehension., Comment: 11 pages, 11 figures, 2 tables. IEEE Vis 2024 Full Paper

The Fermi National Accelerator Laboratory (Fermilab) Linac accepts 750 keV H- ions from the front end and accelerates them to 400 MeV for injection into the Booster rapid cycling synchrotron. Day-to-day drifts in the beam longitudinal trajectory during regular operation are of the order of several degrees. They are believed to cause additional losses in both the Linac and the Booster and are addressed by empirically adjusting cavity phases of front end and Linac RF cavities. This work explores a scheme for expressing these drifts in terms of phase shifts in the low-energy part of the Linac. Such a description allows for a simplified visual representation of the drifts, suggests a clear algorithm for their compensation, and provides a tool for estimating efficiency of such compensation., Comment: 15th International Particle Accelerator Conference (IPAC'24)

Modelling the scrape-off layer of a stellarator is challenging due to the complex magnetic 3D geometry. The here presented study analyses simulations of the scrape-off layer (SOL) of the stellarator Wendelstein 7-X (W7-X) using spatially varying diffusion coefficients for the magnetic standard configuration, extending our previous study. Comparing the EMC3-Eirene simulations with experimental observations, an inconsistency between the strike-line width (SLW) and the upstream parameters was observed. While to match the experimental SLW a particle diffusion coefficient $D \approx 0.2$ is needed, $D \approx 1$ is needed to get experimental separatrix temperatures of 50\,eV at the given experimental heating power. We asses the impact of physically motivated spatially varying transport coeffients. Agreement with experimental data can be improved, but various differences remain. We show that drifts are expected to help overcome the discrepancies and, thus, the development of SOL transport models including drifts is a necessary next step to study the SOL transport of the W7-X stellarator., Comment: 15 pages, 13 figures. arXiv admin note: substantial text overlap with arXiv:2201.06341

We consider diffusion operator $-\Delta + b \cdot \nabla$ in $\mathbb R^d$, $d \geq 3$, with drift $b$ in a large class of locally unbounded vector fields that can have critical-order singularities. Covering the entire range of admissible magnitudes of singularities of $b$ (but excluding the borderline value), we construct a strongly continuous Feller semigroup on the space of continuous functions vanishing at infinity, thus completing a number of results on well-posedness of SDEs with singular drifts. The previous results on Feller semigroups employed strong elliptic gradient bounds and hence required the magnitude of the singularities to be less than a small dimension-dependent constant. Our approach is different and uses De Giorgi's method ran in $L^p$ for $p$ sufficiently large, hence the gain in the assumptions on singular drift. For the critical borderline value of the magnitude of singularities of $b$, we construct a strongly continuous semigroup in a ``critical'' Orlicz space on $\mathbb R^d$ whose local topology is stronger than the local topology of $L^p$ for any $2 \leq p<\infty$ but is slightly weaker than that of $L^\infty$., Comment: Fixed typos

Building on results developed in https://doi.org/10.48550/arXiv.2404.14902, where It\^{o}-SDEs with possibly degenerate and discontinuous dispersion coefficient and measurable drift were analyzed with respect to a given (sub-)invariant measure, we develop here additional elliptic regularity results for PDEs and consider the same equations with some further regularity assumptions on the coefficients to provide a pointwise analysis for every starting point in Euclidean space, $d\ge 2$. Our main result is (weak) well-posedness, i.e. weak existence and uniqueness in law, which we obtain under our main assumption for any locally bounded drift and arbitrary starting point among all solutions that spend zero time at the points of degeneracy of the dispersion coefficient. The points of degeneracy form a $d$-dimensional Lebesgue measure zero set, but may be hit by the weak solutions. Weak existence for arbitrary starting point is obtained under broader assumptions. In particular, in that case the drift does not need to be locally bounded., Comment: 36 pages

In this paper, concerning SDEs with H\"older continuous drifts, which are merely dissipative at infinity, and SDEs with piecewise continuous drifts, we investigate the strong law of large numbers and the central limit theorem for underlying additive functionals and reveal the corresponding rates of convergence. To establish the limit theorems under consideration, the exponentially contractive property of solution processes under the (quasi-)Wasserstein distance plays an indispensable role. In order to achieve such contractive property, which is new and interesting in its own right for SDEs with H\"older continuous drifts or piecewise continuous drifts, the reflection coupling method is employed and meanwhile a sophisticated test function is built.

Solar partially ionized plasma is frequently modeled using single-fluid (1F) or two-fluid (2F) approaches. In the 1F case, charge-neutral interactions are often described through ambipolar diffusion, while the 2F model fully considers charge-neutral drifts. Here, we expand the definition of the ambipolar diffusion coefficient to include inelastic collisions (ion/rec) in two cases: a VAL3C 1D model and a 2F simulations of the Rayleigh-Taylor instability (RTI) in a solar prominence thread based on \cite{PopLukKho2021aa, PopLukKho2021ab}. On one side, we evaluate the relative importance of the inelastic contribution, compared to elastic and charge-exchange collisions. On the other side, we compare the contributions of ion/rec, thermal pressure, viscosity, and magnetic forces to the charge-neutral drift velocity of the turbulent flow of the RTI. Our analysis reveals that the contribution of inelastic collisions to the ambipolar diffusion coefficient is negligible across the chromosphere, allowing the classical definition of this coefficient to be safely used in 1F modeling. However, in the transition region, the contribution of inelastic collisions can become as significant as that of elastic collisions. Furthermore, we ascertain that the thermal pressure force predominantly influences the charge-neutral drifts in the RTI model, surpassing the impact of the magnetic force., Comment: Accepted

Consider the $[0,1]$-valued continuous random field solution $(u_t(x))_{t\geq 0, x\in \mathbb R}$ to the one-dimensional stochastic heat equation \[ \partial_t u_t = \frac{1}{2}\Delta u_t + b(u_t) + \sqrt{u_t(1-u_t)} \dot W, \] where $b(1)\leq 0\leq b(0)$ and $\dot W$ is space-time white noise. In this paper, we establish the weak existence and uniqueness of the above equation for a class of drifts $b(u)$ that may be irregular at the points where the noise coefficient is non-Lipschitz and degenerate, specifically at $u=0$ or $u=1$. This class of drifts includes non-Lipschitz drifts like $b(u) = u^q(1-u)$ for every $q\in (0,1)$, and some discontinuous drifts like $b(u) = \mathbf 1_{(0,1]}(u)-u$. This demonstrates a regularization effect of the multiplicative space-time white noise without the standard assumption that the noise coefficient is Lipschitz and non-degenerate. The method we apply is a further development of a moment duality technique that uses branching-coalescing Brownian motions as the dual particle system. To handle an irregular drift in the above equation, particles in the dual system are allowed to have a number of offspring with infinite expectation, and even an infinite number of offspring with positive probability. We show that, even though the branching mechanism with an infinite number of offspring causes explosions in finite time, immediately after each explosion, the total population comes down from infinity due to the coalescing mechanism. Our results on this dual particle system are of independent interest.

We investigate the interior pointwise $C^{\alpha}$ regularity for weak solutions of elliptic and parabolic equations with divergence-free drifts. For such equations, the integrability condition on the drift can be relaxed and the interior $C^{\alpha}$ regularity for some $0<\alpha<1$ has been obtained previously with the aid of Harnack inequality. In this paper, we prove the interior pointwise $C^{\alpha}$ regularity for any $0<\alpha<1$ provided that the drift is small. We obtain the regularity under three different types conditions on the drift. The proof is based on the energy inequality and the perturbation technique.

In this work, coupled optical and electrical discharge measurements have been implemented to investigate the plasma state of a reactive HiPIMS TiO$_2$ deposition process running at a fixed duty cycle of 2% and at a repetition rate of 1 kHz. Investigations focus on both the effect of the erosion target and substrate-holder temperature in an Ar/O$_2$ gas mixture at fixed working pressure. First, as the racetrack shape evolves with the use of the target, the deposition rate is modified, in the same way as the emission intensity measured. Second, with the heater set at 400 $^\circ$C in pure Ar, the coating appears thinner, and the optical emission spectroscopy measurement reveals the presence of oxygen atoms. Thus, the results from the coupled optical and electrical measurements are sensitive enough to track potential drift of the process.

The ambipolar diffusion approximation is used to model partially ionised plasma dynamics in a single fluid setting. To correctly apply the commonly used version of ambipolar diffusion, a set of criteria should be satisfied including the requirement that the difference in velocity between charges and neutral species (known as drift velocity) is much smaller than the thermal velocity, otherwise the drift velocity will drive a non-negligible level of further collisions between the two species. In this paper we explore the consequence of relaxing this assumption. We show that a new induction equation can be formulated without this assumption. This formulation reduces to the ambipolar induction equation in the case the drift velocity is small. In the large drift velocity limit, the feedback of the drift velocity on the collision frequency results in decreased diffusion of the magnetic field compared with the standard ambipolar diffusion approximation for the same parameters. This has a natural consequence of reducing the frictional heating that can occur. Applying this to results from flux emergence simulations where the expansion of the magnetic field leads to strong adiabatic cooling of the partially ionised chromosphere resulted in a noticeable reduction in the magnitude of the predicted drift velocities., Comment: 12 pages, 4 figures, accepted for publication in PTRSA

In this paper, we consider the time change of the diffusion process on the 2-dimensional unit sphere generated by the Laplace-Beltrami operator, perturbed by means of a longitudinal vector field. First, this is done by addressing the problem of finding strong solutions to suitable time-nonlocal Kolmogorov equations, via a spectral decomposition approach. Next, the desired process is constructed as the composition of the aforementioned diffusion process and the inverse of a subordinator, and it is used to provide a stochastic representation of the solution of the involved time-nonlocal Kolmogorov equation, which in turn leads to the spectral decomposition of its probability density function. A family of operators induced by the process is then adopted to provide very weak solutions of the same time-nonlocal Kolmogorov equation with much less regular initial data. From the spectral decomposition results we also get some bounds on the speed of convergence to the stationary state, proving that the process can be considered an anomalous diffusion. These results improve some known ones in terms of both the presence of a perturbation and the lack of regularity of the initial data., Comment: 32 pages

Continuous machine learning pipelines are common in industrial settings where models are periodically trained on data streams. Unfortunately, concept drifts may occur in data streams where the joint distribution of the data X and label y, P(X, y), changes over time and possibly degrade model accuracy. Existing concept drift adaptation approaches mostly focus on updating the model to the new data possibly using ensemble techniques of previous models and tend to discard the drifted historical data. However, we contend that explicitly utilizing the drifted data together leads to much better model accuracy and propose Quilt, a data-centric framework for identifying and selecting data segments that maximize model accuracy. To address the potential downside of efficiency, Quilt extends existing data subset selection techniques, which can be used to reduce the training data without compromising model accuracy. These techniques cannot be used as is because they only assume virtual drifts where the posterior probabilities P(y|X) are assumed not to change. In contrast, a key challenge in our setup is to also discard undesirable data segments with concept drifts. Quilt thus discards drifted data segments and selects data segment subsets holistically for accurate and efficient model training. The two operations use gradient-based scores, which have little computation overhead. In our experiments, we show that Quilt outperforms state-of-the-art drift adaptation and data selection baselines on synthetic and real datasets., Comment: Accepted to AAAI 2024

The Fermilab Linac accepts the 0.75 MeV H- ions from the front end and accelerates them to 400 MeV for injection into the Booster. Day-to-day drifts of the longitudinal trajectory in the Linac, reconstructed from phase readings of Beam Position Monitors, are at the level of several degrees. They are believed to cause additional losses both in the Linac and Booster, and are addressed by empirically adjusting the phases of Linac cavities. This work explores the option of expressing these drifts in terms of phase shifts in two cavities at the low-energy part of the Linac. Such a description allows for a simplified visual representation of the drifts, suggests a clear algorithm for their compensation, and provides a tool for estimating efficiency of such compensation.

The inclusion of drifts in plasma edge codes like SOLPS-ITER is required to match simulation data with experimental profiles. However, this remains numerically challenging. In this paper, the effect of some numerical factors on the final plasma solution is investigated. This study is performed on three EAST simulations in upper single null configuration: an attached purely deuterium case, an attached case with limited Ne-seeding and a detached Ne-seeded case. The effects of the anomalous conductivity and anomalous thermo-electric coefficient on the plasma potential are investigated. Next, the effect of the employed grids is shown. In order to investigate these effects, accurate drift simulations are needed. Therefore, the employed time step and used numerical parameters are discussed for the three studied simulations. For all presented simulations, it is verified that the restriction of the grid to the first flux surface tangent to the main chamber wall has only a small effect on the divertor solution. This means that the main power dissipation takes place inside the simulated domain and only a small fraction of the power is leaving the B2.5 grid trough the grid boundary closest to the first wall. Finally the effect of drifts on the asymmetry between the inner and outer target for EAST simulations is demonstrated.

In this paper we investigate the existence and uniqueness of weak solutions for kinetic stochastic differential equations with H\"older diffusion and unbounded singular drifts in Kato's class. Moreover, we also establish sharp two-sided estimates for the density of the solution. In particular, the drift $b$ can be in the mixed $L^q_tL^{p_1}_{x_1}L^{p_2}_{x_2}$ space with $\frac2q+\frac{d}{p_1}+\frac{3d}{p_2}<1$. As an application, we show the existence and uniqueness of weak solution to a second order singular interacting particle system in ${\mathbb R}^{d N}$., Comment: 26pages

We quantify superdiffusive transience for a two-dimensional random walk in which the vertical coordinate is a martingale and the horizontal coordinate has a positive drift that is a polynomial function of the individual coordinates and of the present time. We describe how the model was motivated through an heuristic connection to a self-interacting, planar random walk which interacts with its own centre of mass via an excluded-volume mechanism, and is conjectured to be superdiffusive with a scale exponent $3/4$. The self-interacting process originated in discussions with Francis Comets., Comment: 26 pages, 4 figures; v2: minor revision

Let $d\geq 2$. In this paper, we investigate the following stochastic differential equation (SDE) in ${\mathbb R}^d$ driven by Brownian motion $$ {\rm d} X_t=b(t,X_t){\rm d} t+\sqrt{2}{\rm d} W_t, $$ where $b$ belongs to the space ${\mathbb L}_T^q \mathbf{H}_p^\alpha$ with $\alpha \in [-1, 0]$ and $p,q\in[2, \infty]$, which is a distribution-valued and divergence-free vector field. In the subcritical case $\frac dp+\frac 2q<1+\alpha$, we establish the existence and uniqueness of a weak solution to the integral equation: $$ X_t=X_0+\lim_{n\to\infty}\int^t_0b_n(s,X_s){\rm d} s+\sqrt{2} W_t. $$ Here, $b_n:=b*\phi_n$ represents the mollifying approximation, and the limit is taken in the $L^2$-sense. In the critical and supercritical case $1+\alpha\leq\frac dp+\frac 2q<2+\alpha$, assuming the initial distribution has an $L^2$-density, we show the existence of weak solutions and associated almost sure Markov processes. Moreover, under the additional assumption that $b=b_1+b_2+{\rm div} a$, where $b_1\in {\mathbb L}^\infty_T{\mathbf B}^{-1}_{\infty,2}$, $b_2\in {\mathbb L}^2_TL^2$, and $a$ is a bounded antisymmetric matrix-valued function, we establish the convergence of mollifying approximation solutions without the need to subtract a subsequence. To illustrate our results, we provide examples of Gaussian random fields and singular interacting particle systems, including the two-dimensional vortex models., Comment: 40pages

We study a multidimensional stochastic differential equation with additive noise: $$ d X_t=b(t, X_t) dt +d \xi_t, $$ where the drift $b$ is integrable in space and time, and $\xi$ is either a fractional Brownian motion or an $\alpha$-stable process. We show weak existence of solutions to this equation under the optimal condition on integrability indices of $b$, going beyond the subcritical Krylov-R\"ockner (Prodi-Serrin-Ladyzhenskaya) regime. This extends the recent results of Krylov (2020) to the fractional Brownian and L\'evy cases. We also construct a counterexample to demonstrate the optimality of this condition. Our methods are built upon a version of the stochastic sewing lemma of L\^e and the John--Nirenberg inequality.

By using the It\^{o}-Tanaka trick, we prove the unique strong solvability as well as the gradient estimates for stochastic differential equations with irregular drifts in low regularity Lebesgue-H\"{o}lder space $L^q(0,T;{\mathcal C}_b^\alpha({\mathbb R}^d))$ with $\alpha\in(0,1)$ and $q\in (2/(1+\alpha),2$). As applications, we show the unique weak and strong solvability for stochastic transport equations driven by the low regularity drift with $q\in (4/(2+\alpha),2$) as well as the local Lipschitz estimate for stochastic strong solutions.

The goal of this article is to establish a central limit theorem for the Euler-Maruyama scheme approximating multidimensional SDEs with elliptic Brownian diffusion, under very mild regularity requirements on the drift coefficients. When the drift is H\"older continuous, we show that the limiting law of the rescaled fluctuations around the true solution is characterised as the unique solution of a hybrid Young-It\^o differential equation. When the drift has positive Sobolev regularity, this limit is characterised by the solution of a transformed SDE. Our result is an extension of the results of Jacod-Kurtz-Protter (1991, 1998) in which SDEs with differentiable coefficients were considered. To compensate for the lack of regularity of the drifts, we utilize the regularisation effect from the non-degenerate noise.

In this paper, we study McKean-Vlasov SDE living in $\mathbb{R}^d$ in the reversible case without assuming any type of convexity assumptions for confinement or interaction potentials. Kramers' type law for the exit-time from a domain of attraction is established. Namely, in the small-noise regime, the limit in probability of the first exit-time behaves exponentially. This result is established using the large deviations principle as well as improved coupling method. Having removed the convexity assumption, this work is a major improvement of the previously known results for the exit-time problem, the review of which is provided in the paper., Comment: 36 pages, 4 figures

We explore the existence of a continuous marginal law with respect to the Lebesgue measure for each component $(X,Y,Z)$ of the solution to coupled quadratic forward-backward stochastic differential equations (QFBSDEs) {for which the drift coefficient of the forward component is either bounded and measurable or H\"older continuous. Our approach relies on a combination of the existence of a weak {\it decoupling field} (see \cite{Delarue2}), the integration with respect to space time local time (see \cite{Ein2006}), the analysis of the backward Kolmogorov equation associated to the forward component along with an It\^o-Tanaka trick (see \cite{FlanGubiPrio10})}. The framework of this paper is beyond all existing papers on density analysis for Markovian BSDEs and constitutes a major refinement of the existing results. We also derive a comonotonicity theorem for the control variable in this frame and thus extending the works \cite{ChenKulWei05}, \cite{DosDos13}. As applications of our results, we first analyse the regularity of densities of solution to coupled FBSDEs. In the second example, we consider a regime switching term structure interest rate models (see for e.g., \cite{ChenMaYin17}) for which the corresponding FBSDE has discontinuous drift. Our results enables us to: firstly study classical and Malliavin differentiability of the solutions for such models, secondly the existence of density of such solutions. Lastly we consider a pricing and hedging problem of contingent claims on non-tradable underlying, when the dynamic of the latter is given by a regime switching SDE (i.e., the drift coefficient is allowed to be discontinuous). We obtain a representation of the derivative hedge as the weak derivative of the indifference price function, thus extending the result in \cite{ArImDR10}., Comment: 46 pages, New Section with applications was added with a change of title

In this paper, we study a semilinear SPDE with a linear Young drift $du_{t}=Lu_{t}dt+f\left(t, u_{t}\right)dt+\left(G_{t}u_{t}+g_{t}\right)d\eta_{t}+h\left(t, u_{t}\right)dW_{t}$, where $L$ is the generator of an analytical semigroup, $\eta$ is an $\alpha$-H\"older continuous path with $\alpha \in \left(1/2, 1\right)$ and $W$ is a Brownian motion. After establishing through two different approaches the Young convolution integrals for stochastic integrands, we introduce the corresponding definition of mild solutions and continuous mild solutions, and give via a fixed-point argument the existence and uniqueness of the (continuous) mild solution under suitable conditions., Comment: 17 pages

In this article we provide various analytical and numerical methods for calculating the average drift of magnetically trapped particles across field lines in complex geometries, and we compare these methods against each other. To evaluate bounce-integrals, we introduce a generalisation of the trapezoidal rule which is able to circumvent integrable singularities. We contrast this method with more standard quadrature methods in a parabolic magnetic well and find that the computational cost is significantly lower for the trapezoidal method, though at the cost of accuracy. With numerical routines in place, we next investigate conditions on particles which cross the computational boundary, and we find that important differences arise for particles affected by this boundary, which can depend on the specific implementation of the calculation. Finally, we investigate the bounce-averaged drifts in the optimized stellarator NCSX. From investigating the drifts, one can readily deduce important properties, such as what subset of particles can drive trapped-particle modes, and in what regions radial drifts are most deleterious to the stability of such modes., Comment: 12 pages, 6 figures

This work is concerned with homogenization problems for elliptic equations of the type \[ \begin{cases} \mathfrak{L}_{\delta} u_{\delta} + \lambda u_{\delta} = f_{\delta} \qquad \text{in} \;\; D, \\ \qquad \quad \;\, u = 0 \qquad \, \text{on} \;\; \partial D, \end{cases} \] where $\delta > 0$, $\lambda \in \mathbb{R}$, $D$ is a bounded open set in $\mathbb{R}^{d}$, and $f_{\delta} \in H^{-1}(D)$. The operator $ \mathfrak{L}_{\delta} u = -{\rm div} \left( A^\delta \nabla u + C^\delta u \right) + B^\delta \nabla u +k^\delta u $ involved uniformly bounded diffusion coefficients $A^\delta$, where drifts $B^\delta$, $C^\delta$ and potential $k^\delta$ are possibly unbounded. An application to homogenization of the corresponding diffusion processes is also discussed., Comment: 38 pages

We study the quantitative homogenization of linear second order elliptic equations in non-divergence form with highly oscillating periodic diffusion coefficients and with large drifts, in the so-called ``centered'' setting where homogenization occurs and the large drifts contribute to the effective diffusivity. Using the centering condition and the invariant measures associated to the underlying diffusion process, we transform the equation into divergence form with modified diffusion coefficients but without drift. The latter is in the standard setting for which quantitative homogenization results have been developed systematically. An application of those results then yields quantitative estimates, such as the convergence rates and uniform Lipschitz regularity, for equations in non-divergence form with large drifts., Comment: 16 pages